Consider a linear model
$$
y = Ab+n,
$$
where $b \in \mathbb{R^m}$ is a parameter to be estimated, $n \in \mathbb{R^{n}}$ is a noise with mean $\mathbb{E}n = m_{n}$ and with covariation matrix $R_{n}$. The Gauss-Markov estimator is an umbiased estimator of parameter $b$ given by $\beta(y) = B(y-m_{n})$, where $B$ is a solution of an optimisation problem
$$
\mathbb{E}(b - B(y-m_{n}))^2 \to \min\limits_{B}.
$$
Solving this problem we receive
$$
\beta(y) = (A'R_{n}^{-1}A)^{-1}A'R_{n}^{-1}(y-m_{n}).
$$
I'm looking for properties of this estimator. Would it have the minimum variance in the class of unbiased linear estimators? Would it be consistent? And if the answer is negative in general case, maybe it is true if $n$ is a gaussian vector?